Mathematics is not a deductive science - that's a cliché. When you try to prove a theorem, you don't just list the hypotheses, and then start to reason. What you do is trial and error, experimentation, guesswork.

A good stack of examples, as large as possible, is indispensable for a thorough understanding of any concept,and when I want to learn something new, I make it my first job to build one.

Many teachers are concerned about the amount of material they must cover in a course. One cynic suggested a formula: since, he said, students on the average remember only about 40% of what you tell them, the thing to do is to cram into each course 250% of what you hope will stick.

The heart of mathematics consists of concrete examples and concrete problems. Big general theories are usually afterthoughts based on small but profound insights; the insights themselves come from concrete special cases.

It is the duty of all teachers, and of teachers of mathematics in particular, to expose their students to problems much more than to facts.

The most spectacular thing about Johnny [von Neumann] was not his power as a mathematician, which was great, or his insight and his clarity, but his rapidity; he was very, very fast. And like the modern computer, which no longer bothers to retrieve the logarithm of 11 from its memory (but, instead, computes the logarithm of 11 each time it is needed), Johnny didn't bother to remember things. He computed them. You asked him a question, and if he didn't know the answer, he thought for three seconds and would produce and answer.

The only way to learn mathematics is to do mathematics. That tenet is the foundation of the do-it-yourself, Socratic, or Texas method.